Derivative of exponential application

The area of a rectangle at point x is given by:

A = xe-x

The maximum (or minimum) will occur when

`(dA)/(dx)=0`

Now

`(dA)/(dx)=(x)(-e^(-x))+(e^(-x))(1)`

`=e^(-x)(1-x)`

This expression only equals zero when x = 1, and is positive when x < 1 and negative when x > 1, so we have a maximum.

So the maximum area is (1)(e-1) = 0.3679 units2